I almost drove up there from the south when we were in Gallup/ Grants in early August. Glad I didn't now that I've researched the roads from your link.
The water running over I-40 east of Gallup convinced me not to head that far off the normal track. We got six inches of pea sized hail on NM-53 near the Ice Cave and had to stop and wait four hours for it to melt before the road was passable.
I just looked over the NPS web site for
Chaco Culture National Historic Park and noticed this
The campground will go to an online reservation system beginning January 8, 2014. Individual sites will increase to $15/night; group sites will be a flat rate of $60. Senior and Access passes will apply
CampgroundAs far as alternate campgrounds -
Alternate Campgrounds